The enemies encountered are the opposite of your joined faction (obviously). There will be a few points where your NPC assistants will wait while you do the bulk of the work and find ways to advance past certain obstacles. One in particular is a Nordic door that should be familiar to you by now (as seen in other dungeons like The Golden Claw). Here you need the Ebony Claw to progress.

Pick up the Ebony Claw lying near the bodies of the downed enemy soldiers by the door. Examine the Claw in your menu to find out the correct sequence and adjust the inner, middle, and outer rings appropriately to match the pattern on the Claw. Once you do, activate the keyhole and proceed deeper into Korvanjund.

Eventually, you come upon a room where your accompanying NPC tells you to search for the crown. Attack the seemingly lifeless undead on the throne to see it come back to life! This Draugr Scourge will be reanimated along with Wights. Once the Scourge and his friends are defeated, loot the Jagged Crown from the corpse.

It is in fact possible to still switch sides at his point. even if you chose a side you can be sneaky and switch sides by giving the crown to the opposite group that you originally sided with. For example if you are supposed to give the grown to Ulfric you can instead choose to give it to Tullius and then continue with the legion quests as if you started with them.